The Steiner Multicut problem asks, given an undirected graph G, terminals sets T1,...,Tt $\subseteq$ V(G) of size at most p, and an integer k, whether there is a set S of at most k edges or nodes s.t. of each set Ti at least one pair of terminals is in different connected components of G \ S. This problem generalizes several graph cut problems, in particular the Multicut problem (the case p = 2), which is fixed-parameter tractable for the parameter k [Marx and Razgon, Bousquet et al., STOC 2011]. We provide a dichotomy of the parameterized complexity of Steiner Multicut. That is, for any combination of k, t, p, and the treewidth tw(G) as constant, parameter, or unbounded, and for all versions of the problem (edge deletion and node deletion with and without deletable terminals), we prove either that the problem is fixed-parameter tractable or that the problem is hard (W[1]-hard or even (para-)NP-complete). We highlight that: - The edge deletion version of Steiner Multicut is fixed-parameter tractable for the parameter k+t on general graphs (but has no polynomial kernel, even on trees). We present two proofs: one using the randomized contractions technique of Chitnis et al, and one relying on new structural lemmas that decompose the Steiner cut into important separators and minimal s-t cuts. - In contrast, both node deletion versions of Steiner Multicut are W[1]-hard for the parameter k+t on general graphs. - All versions of Steiner Multicut are W[1]-hard for the parameter k, even when p=3 and the graph is a tree plus one node. Hence, the results of Marx and Razgon, and Bousquet et al. do not generalize to Steiner Multicut. Since we allow k, t, p, and tw(G) to be any constants, our characterization includes a dichotomy for Steiner Multicut on trees (for tw(G) = 1), and a polynomial time versus NP-hardness dichotomy (by restricting k,t,p,tw(G) to constant or unbounded).

As submitted to journal. This version also adds a proof of fixed-parameter tractability for parameter k+t using the technique of randomized contractions